Question

The range of value of $\lambda$ for which the circles $x^2+y^2=4$ and $x^2+y^2-4\lambda x+9=0$ have two common tangents is

• A. $\lambda \in \left[\frac{-13}{8},\frac{13}{8}\right]$
• B. $\lambda >\frac{13}{8}$ or $\lambda <\frac{-13}{8}$
• C. $1<\lambda <\frac{13}{8}$
• D. none of these

Answer 1

The correct option is B $\lambda >\frac{13}{8}$ or $\lambda <\frac{-13}{8}$

$C_1(0,0)$ and $r_1=2$
$C_2(2\lambda ,0)$ and $r_2=\sqrt{4{\lambda }^2-9}$
Since, circles have two common tangents
$|r_1-r_2|<C_1{C}_2<|r_1+r_2|$
$|2-\sqrt{4{\lambda }^2-9}|<|2\lambda |<|2+\sqrt{4{\lambda }^2-9}|$
$\Rightarrow$ $\lambda >\frac{13}{8}$ or $\lambda <\frac{-13}{8}$